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Chapter 2: Problem 1

List all partitions of the set \(A=\\{a, b, c\\}\).

Short Answer

Expert verified
The partitions are: \(\{\{a\}, \{b\}, \{c\}\}\), \(\{\{a, b\}, \{c\}\}\), \(\{\{a, c\}, \{b\}\}\), \(\{\{b, c\}, \{a\}\}\), and \(\{\{a, b, c\}\}\).

Step by step solution

01

Understand the concept of a partition

A partition of a set is a way of dividing its elements into non-empty subsets such that every element is included in exactly one subset, and no subset is empty. For any set, the union of these subsets should give back the original set.
02

Identify partitions where each element is in its own subset

The first type of partition is where each element is in its own subset. For the set \(A = \{a, b, c\}\), this partition is: \(\{\{a\}, \{b\}, \{c\}\}\).
03

Identify partitions with one subset containing two elements

Now consider partitions where two elements are grouped together and the third element is alone. The possibilities are:- \(\{\{a, b\}, \{c\}\}\)- \(\{\{a, c\}, \{b\}\}\)- \(\{\{b, c\}, \{a\}\}\)
04

Identify the partition with all elements in one subset

The final type of partition for set \(A\) is where all elements are in the same subset. This partition is: \(\{\{a, b, c\}\}\).
05

List all possible partitions

Combine all these possibilities to list all partitions of set \(A = \{a, b, c\}\):1. \(\{\{a\}, \{b\}, \{c\}\}\)2. \(\{\{a, b\}, \{c\}\}\)3. \(\{\{a, c\}, \{b\}\}\)4. \(\{\{b, c\}, \{a\}\}\)5. \(\{\{a, b, c\}\}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subset
A subset is a selection of elements from a broader set where each element in the subset is also contained in the initial set. This idea is central when discussing partitions, as partitions are made up of subsets.
For example, given the set \( A = \{a, b, c\} \), any collection of elements from \( A \), like \( \{a, b\} \) or \( \{c\} \), is called a subset. It's important to remember that the empty set, \( \{\} \), is also technically a subset of any set, though it isn't used when creating partitions.
When examining subsets for a set partition, each subset must be non-empty, meaning they have to include at least one element.
Some characteristics of subsets:
  • The total number of possible subsets for a set is \(2^n\), where \(n\) is the number of elements in the set.
  • Every set is a subset of itself.
  • Subsets are used to build partitions by dividing all elements of a set into these smaller, non-overlapping groups.
Non-empty subsets
Non-empty subsets are vital for understanding how set partitions work. A non-empty subset is a subset that contains at least one element from the original set.
Within the context of partitions, every subset needs to be non-empty because partitions exclude the possibility of having an empty group. This ensures that every element in the set is included somewhere in the partition.
Consider our example set \( A = \{a, b, c\} \). When forming partitions of \( A \), you'll work exclusively with non-empty subsets like \( \{a\} \), \( \{b, c\} \), or even \( \{a, b, c\} \), but not with empty sets.
  • Each non-empty subset must have at least one element from the original set.
  • No element should be left out across all subsets in the partition.
  • These ensure that the union of all subsets equals the original set.
Union of subsets
When discussing partitions, the union of subsets is a crucial concept. The union is essentially the combination of all elements found in the subsets, and it must reconstruct the original set.
Let鈥檚 take our set \( A = \{a, b, c\} \) as an example. In a partition like \( \{ \{a\}, \{b, c\} \} \), the union of these subsets would return us to the full set \( A \).
The union is concerned with ensuring:
  • All elements of the original set appear, covered across the subset groups.
  • No element gets misplaced; each must belong exactly to one subset in a partition.
  • The combined subsets do not include extra elements not found in the original set.
In doing so, partitions achieve their aim of dividing a set into exclusive, non-empty groups.

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Most popular questions from this chapter

A gardener has three flowering shrubs and four nonflowering shrubs, where all shrubs are distinguishable from one another. He must plant these shrubs in a row using an alternating pattern, that is, a shrub must be of a different type from that on either side. How many ways can he plant these shrubs? If he has to plant these shrubs in a circle using the same pattern, how many ways can he plant this circle? Note that one nonflowering shrub will be left out at the end.

Explain in words why the following equalities are true based on number of subsets, and then verify the equalities using the formula for binomial coefficients. (a) \(\left(\begin{array}{l}n \\ 1\end{array}\right)=n\) (b) \(\left(\begin{array}{c}n \\ k\end{array}\right)=\left(\begin{array}{c}n \\\ n-k\end{array}\right), 0 \leq k \leq n\)

Use the binomial theorem to calculate \(9998^{3}\).

The definition of \(\mathbb{Q}=\\{a / b \mid a, b \in \mathbb{Z}, b \neq 0\\}\) given in Chapter 1 is awkward. If we use the definition to list elements in \(\mathbb{Q}\), we will have duplications such as \(\frac{1}{2}, \frac{-2}{-4}\) and \(\frac{300}{600}\) Try to write a more precise definition of the rational numbers so that there is no duplication of elements.

(a) How many three-digit numbers can be formed from the digits \(1,2,\) 3 if no repetition of digits is allowed? List the three-digit numbers. (b) How many two-digit numbers can be formed if no repetition of digits is allowed? List them. (c) How many two-digit numbers can be obtained if repetition is allowed?
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